Let $G*H$ be the product $*$ of $G$ and $H$. In this paper we determine the rth power of the graph $G*H$ in terms of $G^r, H^r$ and $G^r*H^r$, when $*$ is the join, Cartesian, symmetric difference, disjunctive, composition, skew and corona product. Then we solve the equation $(G*H)^r=G^r*H^r$. We also compute the Wiener index and Wiener polarity index of the skew product.

In future high-speed self-routing photonic networks based on all-optical time division multiplexing (OTDM) it is highly desirable to carry out packet switching, clock recovery and demultplexing in the optical domain in order to avoid the bottleneck due to the optoelectronics conversion. In this paper we propose a self-routing OTDM node structure composed of an all-optical router and demultiplex...

In recent papers, the authors obtained formulas for directional derivatives of all orders, of the immanant and of the m-th ξ-symmetric tensor power of an operator and a matrix, when ξ is a character of the full symmetric group. The operator norm of these derivatives was also calculated. In this paper, similar results are established for generalized matrix functions and for every symmetric tenso...

Consider the algebra Q〈〈x1, x2, . . .〉〉 of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x1, x2, . . .) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In p...

The power index of a square Boolean matrix A is the least integer d such that A is a linear combination of previous nonnegative powers of A. We determine the maximum power indices for the class of n × n primitive symmetric Boolean matrices of trace zero, the class of n × n irreducible nonprimitive symmetric Boolean matrices, and the class of n×n reducible symmetric Boolean matrices of trace zer...

The spatiotemporal nonlinear Schrödinger equation with power-law nonlinearity in PT-symmetric potentials is investigated, and two families of analytical three-dimensional spatiotemporal structure solutions are obtained. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulation. Results indicate that solutions are stable below some thresholds...